ERIC ED387441: Change in Teachers' Conceptions and Practice.

DOCUMENT RESUME

ED 387 441

SP 036 153

AUTHOR
TITLE
PUB DATE
NOTE

PUB TYPE

Fuller, Roberta Ann

Change in Teachers* Conceptions and Practice.
13 Oct 94

30p.; Paper presented at the Meeting of the
Mid-Western Educational Research Association
(Chicago, IL. October 13, 1994). Somewhat blurred
print may cause reproducibility problems.
Reports - Research/Technical (143) -
Speeches/Conference Papers (150)

EDRS PRICE MF01/PC02 Plus Postage.

DESCRIPTORS Case Studies; Concept Formation; Elementary School

Mathematics; Elementary School Teachers; Higher
Educat ion; Inter mediate Grades; -'Mathemat ica 1
Concep*". s ; -'Mathemat ics Instruction; '^'Mathemat i cs
Teachers; '''Reflective Teaching; ''Teacher Improvement;
-''Teacher Role; Teaching Methods

IDENTIFIERS Matheir.a t i cs Education Research; Reflection Process

ABSTRACT

This study examined factors that teachers say
determine whether they modify new information about mathematics,
mathematics learning, and mathematics teaching to fit their existing
conceptions or whether they restructure their existing conceptions.
Three female teachers (two sixth grade and one fifth grade) from a
rural, public school participated in this study. All three teachers
are currently participating in a research project which is
disseminating new information about mathematics, mathematics
learning, and mathematics teaching. There were no specific criteria
used in the selection of the three teachers other than the fact that
they had expressed an interest and willingness to participate in the
study. Each teacher was observed teaching a mathematics lesson, and
an interview was conducted following the observed lesson. Both the
lessons and the interview sessions typically lasted one hour and were
audio-recorded to secure a record for later analysis. Analysis of the
interviews showed that while two teachers' existing conceptions of
mathematics teaching underwent change as a result of their
participation in the research project, one teacher's existing
conceptions of mathematics teaching were preserved, seemingly as a
result of her strongly held conceptions of mathematics and
mathematics learning. The results seem to indicate that, though the
role of the classroom as a learning environment and the role of
reflection may be important to the change process, the role of the
teacher is the most salient, since the teacher exercises considerable
control over the decision of whether or not to implement change.
(Contains 18 references.) (Author/ND)

Reproductions supplied by EDRS are the best that can be made
from the original document.

Change in Teachers' Conceptions and Practice

Roberta Ann Fuller

Illinois State University

Spring 1994

Paper Presented at the
Mid-Western Educational Research Association Conference

Chicago, Illinois
October 13. 1994

Running Head: TEACHERS^ CONCEPTIONS AND PRACTICE

■PERMISSION lO REPHODUCt iMiS 'JL '-ANT vi ^ , of n.nc aiu)'.

Xol^-^

TO THt tOUr.ATiONAi, RFSOuM( r-s
INFOHMAllON CLNTLf? ,[ Mir .

BEST COPY AVAILABLE

Abstract

The purpose of this study was to examine factors teachers say
determine whether they modify new information about mathematics,
mathematics learning, and mathematics teaching to fit their existing
conceptions or whether they restructure their existing conceptions.
Three female teachers (two sixth grade and one fifth grade) from a
rural, public school participated in this study. All three teachers
are currently participating in a research project which is
disseminating new information about mathematics, mathematics learning,
and mathematics teaching. There were no specific criteria used in the
selection of the three teachers other than the fact that they had
expressed an interest and willingness to participate in the study.
Each teacher was observed teaching a mathematics lesson and an
interview was conducted following the observed lesson. Both the
lessons and the interview sessions typically lasted one hour and were
audio-recorded to secure a record for later analysis. Analysis
of the interviews showed that while two teachers' existing
conceptions of mathematics teaching underwent change as a result
of their participation in the research project, one teacher's
existing conceptions of mathematics teaching were preserved
seemingly as a result of her strongly held conceptions of
mathematics and mathematics learning. The results seem to
indicate that, though the role of the classroom as a learning
environment and the role of reflection may be important to the
change process, the role of the teacher is the most salient,
since the teacher exercises considerable control over the
decision of whether or not to implement change.

er|c

Teachers* Conceptions and Practics

1

Introduction

It has been well documented that raost teacher education students,
both preservice and mservice. believe that mathematics consists of
facts, rules, and procedures, that learning mathematics means
remembering the facts, rules, and procedures, and that teaching
mathematics involves telling or showing students the facts, rules, and
procedures (for a list of references see MoDiarmid, 1990). In light
of the current move for reform in mathematics education, many
preservice and inservice teacher education programs usually attempt to
explore, identify, and challenge teacher education students' beliefs
by providing reasonable alternatives. Unfortunately, most teachers
modify the new ideas to fit their existing conceptions instead of
restructuring their existing conceptions (Doyle, 1985; Doyle &
Ponder, 1977; McDiarmid. 1990; Schram, Wilcox, Lanier. Lappan, & Even,
1968; Swans on -Owens, 1985; Grimellini & Pecori, 1968). Thus it seems
that perhaps teacher education needs to address not only the
exploration, identification, and- challenging of beliefs, but also the
factors that contribute to whether or not beliefs are changed.

While both the learning-to-teach literature and the teacher-
change literature seem to suggest that there are several factors which
may contribute to whether teachers modify new information to fit their
existing conceptions or whether they restructure their existing
conceptions, four seem to be crucial to the change process. They are
the role of the teacher (Cobb. Wood, & Yackel, 1990; Hall & Loucks,

Teachers' Conceptions and Practice

2

1978; Lieberman & Miller, 1984; Richardson, 1990; Richardson, 1992;
Wood, Cobb, & Yackel, 1991), the role of the teacher educator or staf f ^
developer (Cobb, Wood, & Yackel, 1990; Richardson, 1992; Wood, Cobb, &
Yackel, 1991), the role of the classroom as a learning environment
(Cobb, Wood, & Yackel, 1990; Doyle & Ponder, 1977; Wood, Cobb, &
Yackel, 1991), and the role of reflection (Anning, 1988; Cobb, Wood, &
Yackel, 1990; Richardson 1990; Schon, 1983; Shulman, 1986; Thompson,
1984).

THE ROLE OF THE TEACHER

The research suggests that teachers exercise considerable control
over the decision of whether and how to implement a change,
Richardson (1990) claims that any change process should both
acknowledge this control, and help teachers understanJi and be held
accountable for the pedagogical and moral implications of their
decisions.

Richardson (1992) discusses a new form of staff development which
is framed in ways of helping teachers themselves explore their beliefs
and knowledge, reconstruct their premises related to teaching and
learning, and alter their practices. She claims that in order for the
teachers to participate in this reconstructive process, they must
acknowledge the power of their own practical reasoning and expertise,
and share in the ownership of the new content that helps them
reconstruct their practical knowledge.

In a landmark study of the middle 1970s (Herman & McLaughlin,

Teachers' Conceptions and Practice

3

1977) important descriptions of the critical importance of
collaboration, teacher participation, arid the practical nature of
school improvement were introduced. This study and others (Herman 3c
McLaughlin, 1977; Lieberman & Miller, 1984; Richardson, 1992; Wood,
Cobb, & Yackel, 1991) have shown that if teachers are involved as
collaborators in both the identification of problem areas and the
search for solutions, and if they have a sense of efficacy about their
own involvement in reform efforts, reforms are more likely to be
implemented and to last.

Similarly, it has been shown that if attempts to get teachers to
change attend to the sorts of concerns teachers have about their owri
practice, they are more likely to be successful (Hall & Loucks, 1978;
Wood, Cobb, & Yackel, 1991; Richardson, 1992). In fact, Cobb, Wood,
and Yackel (1990) claim that teachers must see their current practice
as problematic as a prerequisite mental state necessary for beneficial
collaboration with researchers or staff developers.

The social context that Cobb, Wood, and Yackel (1990) mutually
constructed with their project teacher during the initial sessions of
their study was such that she viewed them as evaluators. In an
attempt to renegotiate the social norms of the relationship, the
project director initiated a dialogue about a topic within the domain
of the teacher* s expertise - her mathematics textbook. The teacher
questioned the director's suggestion that textbook-based instruction
led many children to develop detrimental concepts of place value. The

ERIC

Teachers' Conceptions and Practice

4

teacher referred to the ability of her students to complete textbook
exejf'cises correctly to support her claim that most of them did
understand place value. The project director suggested that she
conduct her own interviews with some of her students to determine
whether his claims were viable or not. In the course of the
interviews, she began to realize that even though she had carefully
taught her students the algorithmic procedures specified in the
lextbook and although they could produce correct answers, they did not
truly understand place value. Cobb, Wood, and Yackel state that their
genuine collaboration with the project teacher begarj when she realized
That her current instructional practices were problematic,
THE ROLE OF THE TEACHER EDUCATOR OR STAFF DEVELOPER

Richardson (1992) claims that the staff developer requires
extensive knowledge of the formal content and a manner that is self-
effacing. Most importantly, she claims that the staff developer must
help the teacher participants to redefine the content of the staff
development process to include their own practical knowledge as
equally legitimate to the development of shared meaning as the staff
developers' formal knowledge. Thus the staff developer plays a
critical role in the process of creating and' maintaining a
construct ivist and empowering process that has a specific content as
its focus.

Cobb. Wood, and Yackel (1990) further claim that the staff
developer's role is to help the teacher "develop personal,

Teachers' ConceFticns and Practice

5

experiential ly-based reasons and motivations for reorganizing
classroom practice" (p. 144) rather than to show the teacher how to
teach in a specified way. As a result, they do not directly try to
change the manner in which teachers themselves teach. Instead, they
encourage them to make their practice compatible with a constructivist
view of the nature of mathematical activity and learning.

Wood, Cobb. & Yackel (1991) claim that if teachers are going to
make significant changes in their ways of teaching, they will need
continued support as they encounter dilemma.'s and conflicts and that
this means finding ways to guide and support teachers as they learn in
the setting of their classrooms.

THE ROLE OF THE CLASSROOM AS A LEARNING ENVIRONMENT

Cobb, Wood, & Yackel 's (1990) work with teachers is based on the
assumption that beliefs and practice are dialect! cally related.
Beliefs are expressed in practice, and problems or surprises
encountered in practice give rise to opportunities to reorganize
beliefs. When analyzing the project teacher's learning, Cobb, Wood,
and Yackel argued that her beliefs and practices were interdependent
and developed together. They claim that it is precisely because of
this interdependency that her classroom was her primary learning
environment.

Cobb, Wood, and Yackel began to realize that researchers
construct formal models in contexts that are incompatible with those
in which teachers construct the knowledge that informs their practice.

ERIC

T^faohers' Conceptions and Practice

6

Formal models are a product of a series of abstractions and
formalizations made by researchers who operate in the context of
academic reasoning and attempt to satisfy the current standards of
their research community. In contrast, teachers operate in the
context of pragmatic pedagogical problem solving in which they have to
make on the spot decisions as they interact with their students in
specific situations.

This context is what Doyle and Ponder (1977) refer to as the
practicality ethic. The essential features of this ethic are
f:;ummarised as follows. "Teachers receive a variety of messages
intended to modify and improve their performance. If one listens
carefully to the way teachers talk about these messages, it soon
becomes clear that the term 'practical' is used frequently and
consistently to label statements about classroom practices. The
labeling represents an evaluative process which is a central
ingredient in the initial decision teachers make regarding the
implementation of a proposed change in classroom procedure. Messages
which are seen as practical will be , incorporated into teacher plans,"
(p. 2) The study of the practicality ethic, then, is the study of
perceived attributes of messages and the way in which these
perceptions determine the extent to which teachers will attempt to
modify classroom practices.

To qualify minimally as practical, a change proposal must
describe a procedure in terms which depict classroom contingencies.

Teachers' Conceptions and Practice

7

This alone, however, does not determine practicality. Teachers also
make decisions in terras of the extent to which a proposed procedure is
congruent with perceptions of their own situations. The final
criterion of practicality is described by Doyle and Ponder (1977) as
cost. It refers primarily to the ease with which a procedure can be
impleirtented and the potential return for adopting an innovation.
THE ROLE OF REFLECTION

Recall that Wood, Cobb, & Yackel (1991) examined a teacher's
learning in the setting of the classroom. The teacher changed in her
beliefs about learning, teaching, and the nature of mathematics and
developed a form of practice compatible with constructivism. These
alterations occurred as she reflected on and resolved conflicts and
dilemmas that arose between her previously established form of
practice and the emphasis of the project on children's construction of
mathematical meaning.

Schon (1982), Shulraan (1986), and Anning (1988) claim that
classroom experience is educative only with reflection and that this
•^iuggests that the improvement of the teacher-learning process requires
acknowledging and building upon teachers' experiences, and promoting
reflection on those experiences.

Richardson (1990) further claims that taking control of one's
justifications involves reflection on practices, that is on activities
and their theoretical frameworks, and an ability to articulate them to
others in a meaningful way. A new classroom activity should be

ERIC

TeacherB' Conceptions ana fractice

8

introduced to teachers with an opportunity for them to relate the
activity's theoretical framework to their own beliefs and
understandings. She claims that empowerment is threatened when
teachers are asked to make changes in activities without being asked
to examine their theoretical frameworks, and that in fact, teacher
empowerment does not occur without reflection and the development of
the means to express justifications.

Improvement of the teacher-learning process may also require
promoting reflection on beliefs. Thompson <1984) found that
differences in teachers' awareness of the relationships between their
beliefs and their practice seemed to be related directly to
differences in their reflectiveness - in their tendency to think about
their actions in relation to their beliefs. As a result of a failure
to reflect on actions in relation to beliefs, and in the face of other
pressures, beliefs seem to have little effect on teaching.

From this literature, it appears that factors to be considered ir.
determining whether teachers modify new information to fit their
existing conceptions or whether they restructure their existing
conceptions might include the role of the teacher, the role of teacher
educator or staff developer, the role of the classroom as a learning
environment, and the role of reflection. However, these have not been
studied directly, but have simply been discussed as characteristics of
successful staff development programs. The teachers involved were
never interviewed.

ERIC

11

9

The purpose of this study was to examine factors teachers say
determine whether they modify new information about mathematics,
mathematics learning, and mathematics teaching to fit their existing
conceptions or whether they restructure their existing conceptions.

Method

The method of inquiry used was the grounded theory method (see
Strauss & Corbin, 1990). Three middle school teachers participated in
the study. Each teacher was observed teaching a mathematics lesson
and an interview was conducted following the observed lesson. There
was no time overlap among the case studies - no two were conducted
simultaneously.

There were several reasons for observing the teachers prior to
interviewing them. One reason was to become better acquainted with
the sociax context before starting the more direct inquiry in the
interviews. Another reason was to generate conjectures about what the
teacher's conceptions might be, and thus gain a better sense of
direction for later probing. This procedure allowed for inferences
that led to a tentative characterization of the teacher's conceptions
based only on her instruction, without direct input concerning her
professed beliefs and views, and was intended to avoid the potential
influence that the teacher's professed views might have on the
Investigator's sensitivity to the different events observed.

Both the lessons and the interview sessions typically lasted one
hour and were audio- recorded to secure a record for later analysis.

10

THE TEACHERS

Three female teachers (two sixth grade and one fifth grade) from
a rural, public school participated in this study. All three teachers
are currently participating in a. research project which is
disseminating new information about mathematics, mathematics learning,
and mathematics teaching. There were no specific criteria used in the
selection of the three teachers other than the fact that they had
expressed an interest and willingness to participate in the study.

The three teachers were Rhonda, Patricia, and Carla. They had
been teaching for four, seven, and eight years respectively at the
same school.

The Case Studies

What follows is a discussion of the relationships denoting causal
conditions for change in or preservation of each teacher's existing
conceptions of mathematics teaching, the corresponding change in or
preservation of their existing teaching practices, any intervening
conditions, and consequences.
RHONDA

Rhonda's interview revealed three areas v/here her existing
conceptions of mathematics teaching underwent change as a result of
her participation in the research project. They were the use of the
textbook, the use of algorithms vs. discovery, and the focus on
students' thinking vs. answers.

Teachers' Conceptions and Practice

11

Rhonda has decided that she does not want to refer to her book as
much as she did prior to participating in the project because the book
•'shows students exactly how to do it and they would just look in the
book and do it the way the bock did it and I don't want that. ^ She
now uses the book only for writing objectives and uses story problems
to teach concepts. Two intervening conditions facilitated this
response: a lack of direction (she "didn't know where to start" when
she first attempted to implement ideas from the research project) and
the perception that using the book and doing story problems were so
unrelated that she could not do both.

In fact, Rhonda claims that "the big practice change is just
going from doing number sentences to story problems." To do this she
lets "the kids make up the story problems" or lets them choose a topic
and she makes up a problem around that topic. Two intervening
conditions facilitated this strategy: the project provided a "sheet
that gives samples of different types of story problems" which meikes
"it really easy to make up story problems on the spur of the moment"
and Rhonda finds it "helpful, very helpful, when we meet with our
crrade levels because you can hear what other kids are thinking, what
other teachers have tried because sometimes I have a hard time coming
up with enough story problems." Consequently, it takes longer for
Rhonda to plan for mathematics since she has to make up her own tests
and her own worksheets "rather than using the book's numb":

ERIC

Teachers' Conceptions and Practice

12

sentences. " but she claims that she has "covered" over twice as much
material as she did when she used the book and she believes that her
students are better "prepared for real life situations."

Two intervening conditions constrained Rhonda's change in
teaching practice and she responded by modifying the new information.
Rhonda is "kind of worried" about her students' performance on the IGA?
(Illinois Goals Assessment Program) this year for a variety of reasons
("these testings stay in their file until they graduate from high
school." "it's going to influence their next year's teacher, because
if they all receive low scores in math, she's going to adapt her
program for low level kids," and if "the school's rankings are
reported way down in math, somebody's going to be in trouble"). For
this reason and because "the kids and their parents don't feel
comfortable with them not knowing number sentences. " she states that
"what I have them do is, after they do the story problem, and I feel
like they've gotten an understanding of it, I will say, 'Can you write
me a number sentence that could go along with that?' Then when
they've written whatever the number sentence is, I'll say. 'That is
actually what you just did, so if you just see it as a number sentence
somewhere, you'll have a place to start." Sh© claims that "I've kind
cf adapted that because I felt it would be necessary for both the kids
and myself. "

One intervening condition not only constrained Rhonda's change in
teaching practice, but also served as a causal condition f - -

ErJc ')

Teachers' Conceptions and Practice

13

preserving her existing conceptions about the use of the textbook.
Because Rhonda is "having a hard time coming up with a metrics unit
without going back to the book, she states that she "may refer to it
quite a bit more during that time," especially since "the book does
not have a bad metrics unit. "
Tb£L liJSie Alggrithms YSL^ Discovery

Rhonda is now convinced "that there is more than one way to solve
problems, that the algorithm is one way, not necessarily the best way,
and that students must come up with their own different ways of
thinking, " because the project has shown videos of children solving
problems using invented strategies and this has given Rhonda "the idea
that if we teach a strategy, that sometimes can hurt the child and
that children can discover if we let them. " As a result, Rhonda now
lets her students discover their owii way of doing the mathematics and
has them share their ideas with one another. Several intervening
conditions facilitated this response: the project began to
concentrate more on middle school topics and how to permit discovery
at that level, Rhonda "actually got in her classroom and started
trying it" with a positive reaction from the students, she therefore
began to feel more "comfortable with it, "' and Rhonda "found out that
kids can learn how to do it in their heads just as easily, and as a
matter of fact more efficiently than writing it down on paper most of
the time. " According to Rhonda, consequences include students being
convinced that there is more than one way to solve a problem, students

ERIC

Teachers' Conceptions and Practice

14

understanding what they are doing, and students being "able to go back
to their own way of solving, because letting them discover how to
solve it on their own will stick with them. "

On the other hand, two intervening conditions constrained
Rhonda's change in teaching practice and she responded by modifying
the new information. Since Rhonda's school lacks the funds for
manipulative materials, she has had to "come up with a different way"
to enhance the discovery process, usually by having the students draw
pictures of the manipulatives. In addition, Rhonda found that when
several "kids share different ways, the other kids get bored, " so she
decided to use small groups instead of the whole class for sharing
ideas .

One set of intervening conditions not only constrained Rhonda's
change in teaching practice, but in fact served as a causal condition
for preserving her existing conceptions about the use of algorithms.
Rhonda struggles with "letting them discover strategies at the sixth
grade level because they've already been taught a lot of algorithms
and strategies and since this is the first year they've had anything
to do with this program, students who have no number skills have
become frustrated at times where they couldn't discover a way and had
no place to even start. " In this case, Rhonda tries "to give the
student a place to start" and sometimes she wants "to give them
sorr3thing that they can use, some way that they can solve it, and
that's usually the algorithm."

IV

TeacLiers* Conceptions and Practice

15

IhSL Fqsus on Students ^ Ibinkiii^ Answers

Rhonda is convinced that teachers should focus on students'
thinking in addition to the answers that they give to problems because
the videos of children solving problems also "showed us how kids were
thinking, which was wonderful,** Therefore, Rhonda has "worked out a
system" where she gives them "partial credit if they've set everything
up and they're going toward the right direction, but maybe they made
an addition error. " The following is another strategy that Rhonda
uses. "When we work in class, if we did a- story problem, I'd say,
'How did you solve it? Did anybody solve it a different way?' And
they explain until we've exhausted the number of ways that the kids
solved it. And then I may say on the next problem, 'You can't solve
it the same way you solved this one. Choose a different way."

According to Rhonda, consequences include students being able to
"explain a problem, " students "having to think about what they did, "
students understanding one anothers' solutions, teachers "seeing that
students really understand what they're doing," students being helped
to "do more in their heads than on paper, " students becoming
"confident in their own thinking," and teachers "learning from the
kids, because a kid may come up with a way" that the teacher has never
thought about before.
CARLA

Carla's interview also revealed three areas where her existing
conceptions of mathematics teaching were changed as a result of her

ERLC

16

Teachers* Conceptions and Practice

16

participation in the research project. They too were the use of the
textbook, the use of algorithms vs. discovery, and the focus on
students' thinking vs. answers.
Ih^ lias Ql ihfi Textbook

At the beginning of the school year, following the project* s
summer seminar, Carla decided that she did not want to use the book as
often as she did prior to participating in the project, because she
"felt [that] what the program [was] trying to get across was a good
idea. " She claims that it "does make sense. We just teach out of the
book and we expect [the students] to know it and if they don't, well
we work with them and work with them until we think they know it and
some kids don't and so they just get pushed to the side." At that
time, Carla primarily used the book "to pull out the concepts." Three
intervening conditions facilitated this response: "being able to tall,
to other teachers when [she] got with [her grade level] group, "
"having Rhonda right next door who's in the project," and the
realisation that "there's other resources out there to help you . . .
not just . . . the book. "

Nevertheless, several intervening conditions not only constrained
Carla' s change in teaching practice, but also served as causal
conditions for preserving her existing conceptions about the use of
the textbook. According to Carla, "it takes a lot more time" to adapt
to this new way of thinking than what she thought at the beginning.
She "think[s] that the teacher just need[s] time to organize [his or

ERIC

Teachers' Conceptions and Practice

17

her] thoughts and to plan. " Carla is also "concernCed] . . about next
year when [her students] have to go flying back into the book."
Finally, but perhaps most importantly, Carla states that "it was very
difficult," not only for her but for her students as well, because
"students, especially at sixth grade who have really stayed in the
book for the last five years, . . . rely on more structure than this
type of project [provides]." In fact, Carla claims that her students
"can't handle that loose structure" and that even she "can't handle
it. " As a result, Carla has "since the beginning of the year gone
back in the book a lot more, " because her students "feel more
comfortable with it [and she] feeiCs] more comfortable." Carla claims
that her students "really like that feeling that they can bring their
book and open it up and have something there." She "think[s3 that
using their textbooks is a big security blanket for them. " When askec
specifically what influenced her to decide to return to her previous
practice, Carla stated,

because I wasn't comfortable standing up in front of the class
all day long doing problem solving. Especially at sixth grade.
They're not going to want me up there for an hour every day the
whole time doing problem solving. And also because I feel that
they wanted a worksheet on something. The students themselves.
Just their reactions to the problem solving and when I would give
them a worksheet they would feel relieved. They felt like it was
something that they could do even though they might not still

Teachers' Conceptions and Practice

18

understand it.
Ihfi U&& q1 Algorithms vs. Dipcpverv

Rather than simply "teaching [an] algorithm" to her students,
Carla now "want[s] them to try to come up with [it] ... to find it
out themselves, *' because

the results that [the project staff has] shown us . . . and
things that they've talked about , , . make sense, I felt the
idea behind it was . . , in the long run it's probably what we've
always been trying to achieve but yet we just never knew exactly
• . . It wasn^t put in front of us that this is the ultimate goal
we're trying to achieve with the students, . . . that they really
understand what math is or how to go about getting these problems
. . , that there's a reasoning behind them . . • [that] it's not
just because they were taught "this is the way you do it. "
As a result, ' Carla has "start[ed] . . . bringing in the manipulative?.'
drawing pictures, and doing different things like that," though
several intervening conditions constrained this response. First, in
"trying to use [discovery] in the classroom" with her students, Carla
has found that it is "hard all of a sudden just to stop what they're
doing and start going back to the basics," because "at this level [the
students] haven't been exposed to this." Carla claims that "they . .
think . . . that way . . . when they're at first grade until they
start learning the algorithm [and] then they start relying on it and
by the time they're at sixth grade they know that there's an algorithm

/C 1

Teachers* Conceptions and Practice

19

to do all of these problemf?. " Second, Car la is "worried about next
year, " She "wonder[s] . . . how much reteaching [of] the algorithmfsl

. . [the] math teacher [is] going to have to do to satisfy her so
that the students , , . [can] do [the] types of problems , , , in the
book." Finally, Carla has "had a hard time finding the right groups
that work well together," since "several students , . . rely on
everybody else to do all their work for them." However, as a
consequence of this change in teaching practice, Carla claims that "a
majority" of her students would say that "doing mathematics • , . is
solving problems . . . finding different ways to solve problems, " as
opposed to "doing . . . algorithm type worksheets or pages out of the
book. "

Has FoOMg on Students \ Thinking vs. Answers

As a result of her participation in the research project, Carla

is now convinced that it is important to allow students an opportunity

"to share [the] ideas that [they] have in solving problems." To

illustrate this she relates the following:

You know last year when a member of the project staff came to
observe me, one of my students said, "I can solve this problem a
different way." And I said, "Okay, well we don't have time to
discuss that today. " This year I would say, "Okay, what is it?"

PATRICIA

Patricia's interview revealed two areas where her existing
conceptions of mathematics teaching were preserved seemingly as a

22

Teachers' Conceptions and Practice

20

result of her strongly held conceptions of mathematics and mathematics
learning. They were the use of algorithms vs. discovery and the focus
on students' thinking vs. answers.
Ihfi UlLS. Algorithms vs. Discovery

Patricia believes that knowing how to do a procedure is important
but understanding the procedure is not. Therefore, she emphatically
believes that "we do have to teach the algorithms" and is opposed to
using discovery. One reason for this opposition is that if students
know how to use the algorithm, teachers do not Qfissi to use discovery,
as can be seen by the following comments:

The project started out Kindergarten through third grade. Now
they've expanded it to fourth through sixth grade. And we have
really found a lot of frustration in that because we are getting
kids who have come out of a very traditional program and now they
come to fifth gx ide and want to start at fifth grade level type
probleus and go back and draw a picture of this. The kid goes,
"What for? I write the number sentence. I know how to add. I
know how to subtract. Why should I draw a picture?" And the
idea is well, do they really understand place value. Well, maybe
they don't. I don't know if I really understand place value.

We were working the other day in the class that we went to. We
had a fraction problem. Well I could figure that. I just either
use the reciprocal or invert and multiply. And I don't know why

Teachers' Conceptions and Practice

2'i

invert and multiply, but it works. I don't really care. So then
pretty soon, okay, let's take memipulatives and show this. And
we were sitting there figuring and fiddling with it and pretty
soon I said, wait a minute. We are not showing this problem with
manipulatives. We're trying to manipulate the stuff to make the
answer we know is right work. We're not helping our
understanding of the problem. It was backwards. The
manipulative was not helping us to solve the problem. We already
had it solved.

A second reason for this opposition to discovery is that Patricia
believes that "there are people that survive in this world and do very
well and never have a good number sense because a lot of it is an
innate ability." In this case, she believes that teachers shouid not
use discovery because "if you're dealing with somebody who has a
number sense problem to begin with, throwing out a bunch of stuff
the kid becomes so involved in the process of manipulating, they
forget what they're supposed to do, it's just a garae. "

In addition, Patricia believes that discovery Qoanotc Jafi
successful arid in fact, ia noi successful. She believes that "while
they're sitting in your classroom, you cannot give a kid day to day.
living experiences" like those "outside of the classroom, where they
have to use it." "Wl-ien we do these discovery tasks, they don't get
it." In other words, according to Patricia, students do not make the
connection between th< discovery task and the concept being taught.

24

Teachers' Conceptions and Practice

22

The Focus on S tudents ' Thinking va. Answera

Patricia believes that answers are important but solutions are
not. Therefore, Patricia believes that "we should use standards of
representation that are readily acceptable." One reason for this
belief is that "math is one of the things that can be a little bit
concrete, the answer's right or wrong."

A second set of reasons for this belief about mathematics
teaching has to do with Patricia's beliefs about the learner, as is
evidenced by the following:

My kids are very much cued into answers because I think we have
been ... we have programmed our kids to find the answei .
The answer's right or wrong and I think our kids think that way

because we think that way.
Quite often students can't really tell you what they did and they

certainly can't make that make sense to someone else.
If everybody in the group got the same answer, you can pretty

much bet they all solved it the same way, because they have
their mathematical algorithms up here.
Fifth graders are not interested in listening to everybody else.
They do not want to hear the thinking process of 26 kids on
one problem. They do not have the patience for that. Most
times as an adult we don't know how the other person found
that answer. And we don' o really care.
In addition, Patricia believes that "drawing out 101 beautiful

Teachers' Conceptions and Practice

23

pictures to answer a problem" is not efficient, communicative, or
worthwhile.

Discussion

It appears that because Rhonda and Carla viewed their previous
practice as problematic (Cobb, Wood, St, Yackel, 1990), they were able
to engage in the identification of problem areas and the search for
solutions (Berman & McLaughlin, 1977; Lieberman 8c Miller, 1984;
Richardson, 1992; Wood, Cobb, & Yackel, 1991), thereby sharing in the
ownership of the new content (Richardson, 1992). Though various
components of project meetings were mentioned as motivations for
changing conceptions and/or practice, the researcher's role in this
process was not specifically addressed. For two of the three teacher-
identified areas of change in conceptions of mathematics teaching,
Rhonda and Carla both regarded the corresponding change in practice as
•■practical" (Doyle & Ponder, 1977), made the decision to implement the
change, and therefore learned in the setting of their classrooms
(Wood, Cobb, 5c Yackel, 1991). This is also true for Rhonda in
relation to the proposed change in the use of the textbook. However,
since this change did not fit with Carla' s perceptions of her own
situation (Doyle & Ponder. 1977), it was too difficult for her to
implement. Perhaps this occurred only in this instance ber^ause,
unlike Rhonda, Carla did not reflect on and resolve conflicts and
dilemmas that arose between her previously established form of
practice and the proposed change (Wood, Cobb, & Yackel, 1991). This

ERIC

Teachers' Conceptions and Practice

24

lack of reflection seems to have been due to time constraints.

Conversely, it appears that because Patricia does not view her
current practice as problematic (Cobb, Wood, & Yackel, 1990), her
existing conceptions of mathematics teaching have not changed. When
asked what it would take for teachers to adapt to the pedagogy
advocated by the project, she responded, "First of all it would take a
commitment and a ^esire and it would also take the decision that it's
necessary to change. And I'm not sure personally that it is."

Together these results seem to indicate that, though the role of
the classroom as a learning environment and the role of reflection may
be important to the change process, the role of the teacher is the
most salient.

Recommendations for Future Research
Although this study provides additional insights as to why
teachers modify new information about mathematics, mathematics
learning, and mathematics teaching to fit their existing conceptions
or change their existing conceptions, it raises other questions, my
are some teachers more resistant to change than others, as in the case
of Patricia? How do personality differences contribute to this
resistance? How do differences in knowledge, whether it be content
knowledge, pedagogical knowledge, or pedagogical content knowledge
(Shulman, 1986), affect resistance to change? And finally, how do
teachers' beliefs, views, and preferences relate to the change
process?

2V

> *

Teachers' Conceptions and Practice

25

References

Anning, A. (1988). Teachers' theories about children's learning. In J.
Calderhead (Ed.) laagharal Professional Lfiamiag. (pp. 128-145).
New York: Falmer.

Berman. P. 5c McLaughlin. M. (1977). Egdgral Erflgrams Support ^inff

EduQa tl on f^l Chanae^ mi E^iitara Affection Impiem^nt.tinn
aad Cgntinuity. Santa Monica: Rand Corporation.

Cobb. P.. Wood. T. & Yackel. E. (1990). Classrooms as learning

environments for teachers and researcher. In R. Davis, C. Maher &
N. Noddings (Eds. ) Congtruot 1 Vi f?t Views en JthS Teaching aad
Learning Mathfimatics. (pp. 125-146). Reston. VA: National
Council of Teachers of Mathematics.

Doyle, w. (1985). .Qonteni BfiEHfismtatiau ia ZsacJafiral Definitinn.
AQad£Kii<2 Horis. Austin: Research and Development Center for
Teacher Education. University of Texas at Austin.

Doyle. W. & Ponder. G. (1977). The practicality ethic in teacher
decision-making. Interchftni^fj, 8(3). 1-12.

Gritnellini Tomasini, N. & Pecori Balandi. B. (1988). ConaffiBiiuai Ctiangfi
ana Scifiacfi leaciier Tra inin g: A Construr.tivi.^t Perspent.iv^
Paper presented at the annual meeting of the American Educational
Research Association, New Orleans.

Hall. G. & Loucks. S. (1978). Teacher concerns as a basis for
facilitating and personalizing staff development. Teachern
ilfilLsgfi fiacord. 80(i), 36-53.

ERIC

2 b

Teachers' Conceptions and Practice

26

Leiberman, A. & Miller. L. (1984). School improvement: Themes and

variations. Teaghers Qsillftis. Bficord, 86(1). 4-19.
McDiarraid. G. (1990). Challenging prospective teachers' beliefs during

early field experience: A quixotic undertaking? Journal

T'SaQher Education. 41(3), 12-20.
Richardson, V. (1990). Significant and worthwhile change in teaching

practice. Educational Researcher . 19(7), 10-18.
Richardson. V. (1992). The agenda- setting dilemma in ei constructivist

staff development process. Teaching 4 Teacher Education .

8(3). 287-300.

Schon, D. (1983). Jhs. Rgnectivg Practitioner . New York: Basic Books.

Schram, P., Wilcox, S. , Lanier. P., Lappan, G.. & Even, R. (1938).
Chanf?ing Mathematical Conceptions £rf Preservica Teachers : ^
Contgnt an^ PgdSi^Qgigal Intervention . Paper presented at the
annual meeting of the American Educational Research Association,
New Orleans.

Shulman, L. (1986). Those who understand: Knowledge growth in

teaching. Eduoaticnai P.esearcher. 1.5, 4-21.
Straus, A. & Corbin, J. (1990). EaaiSfi Ql Qualitative Research.

Newbury Park: Sage.
Swanson-Owens, D. ( 1986). Identifying Natural Sources q£. Resistance:

A ilaas Siudy Analysis oi Currigulum implementation. Paper

presented at the annual meeting of the American Educational

Research Association, Chicago.

ERIC

Teachers' Conceptions and Practice

27

Thompson, A. (1984). The relationship of teachers' conceptiona of

mathematics and mathematics teaching to instructional practice.

EdUQatj.Qnal StudiSS in Mathematics . 15. 105-127.
Wood. T.. Cobb, P. & Yackel, E. (1991). Change in teaching

mathematics: A Case study. American Educational Researnh Journal .

28(3), 587-616.

o St'

ERIC